Testing Properties of Distributions over Big Domains

We describe an emerging research direction regarding the complexity of testing global properties of discrete distributions, when given access to only a few samples from the distribution. Such properties might include testing if two distributions have small statistical distance, testing various independence properties, testing whether a distribution has a specific shape (such as monotone decreasing, k-modal, k-histogram, monotone hazard rate,...), and approximating the entropy. We describe bounds for such testing problems whose sample complexities are sublinear in the size of the support.